Conjugate gradient methods for red/black systems on vector computers

This research concerns the development, analysis, and comparison of certain conjugate gradient (CG) methods for the solution of generalized Poisson equations $\nabla(K \nabla u) = f$ where K and f are functions of the three spatial variables. We discretize by finite differences with variable spacing and, by diagonal scaling we obtain a symmetric positive definite linear system of equations. The primary method used is the SSOR polynomial preconditioned conjugate gradient (PPCG) method. A red/black ordering is used so that the SSOR iteration may be vectorized. We consider a scaling of the main diagonal of the coefficient matrix to the identity and incorporate the modification of Eisenstat (1981) into the SSOR preconditioner. We also consider the cyclic and reduced system CG (CCG and RSCG) methods where computation is saved by working with vectors of only half-length. We show that, for red/black systems, $\omega\sb{opt}$ = 1, in the sense of minimizing the condition number, for the m-step SSOR PCG method and discuss the choice of $\omega$ for the m-step SSOR PPCG method. Although the diagonal scaling greatly reduces the iterations for the CG method, we show that it has no effect, in terms of the condition number, when preconditioning is employed. It is known that the SSOR PCG method cannot be fully vectorized in the natural ordering although the method will converge in fewer iterations if this ordering is used. We introduce some new orderings based on the red/black ordering to try to obtain at least a partial vectorization of the SSOR iteration while maintaining the superior rate of convergence of SSOR in the natural ordering.